Starting with a single stick of unit length, a point $p \in (0, 1)$ is picked uniformly at random along the stick and the stick is snapped, producing two sticks of length $p$ and $1-p$.

At each next stage, a stick is picked uniformly at random, and a point is picked uniformly at random along the length of that stick, and it is snapped.

Question: After n snaps, what is the expected length of the longest remaining stick?

Remarks:

Myself and a friend of mine did some simulations and found some pretty unexpected results. The expected value after $500$ splits is approximately $0.2098$, which is massive for that many splits, at least intuitively.

On the other hand, it can be proven rather easily that the expected value does go to $0$ as $n \to \infty$. But the decay seems extremely slow.